Foundations — Liquid state — vapour pressure, viscosity, surface tension

Before you can read a single equation in the parent note, you must own the alphabet it is written in. This page introduces every symbol, quantity, and tool in the order they depend on each other. Nothing is used before it is built.

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0. The two players in every equation

Everything reduces to two competing quantities. Meet them first — Figure s01 shows the contest directly: the magenta arrows are the IMF pulling molecules together, the orange arrows are thermal jiggle trying to fling them apart. Keep this picture in mind for the entire page.

Figure s01 — Two molecules (violet) caught between inward IMF pull (magenta) and outward thermal jiggle (orange). Every property in this chapter scores who wins.

The whole chapter is: stickiness vs jiggle. Keep that mental image.

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1. Temperature $T$ and the Kelvin scale

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2. Force $F$, area $A$, and pressure $p$

Figure s02 — Same force $F$, two contact areas. Large area $A$ → small pressure; tiny area $A$ → big pressure. This is $p=F/A$ made visible.

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3. The Boltzmann idea — who has enough energy to escape

The molecules do not all carry the same energy. Some crawl, some sprint.

Here two new symbols appear — earn them now.

Figure s03 — $e^x$ (magenta) explodes for a positive exponent; $e^{-x}$ (violet) decays for a negative one. The exponent $x$ plays the role of $E/RT$: raise $T$ (shrink $E/RT$) and more molecules clear the barrier.

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4. Energies of change — $\Delta H_{vap}$, $E_a$, and $\Delta G$

The symbol $\Delta$ (Greek "delta") always means change in = (final − initial).

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5. Geometry symbols — length, area, volume, radius, angle

Surface tension and capillary rise are geometric, so nail these shapes.

Figure s04 — Left: water, $\theta <90^{\circ }$, concave surface, rises. Right: mercury, $\theta >90^{\circ }$, convex surface, dips. The boundary $\theta =90^{\circ }$ (flat surface) would give exactly zero rise.

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6. Rates of change — the derivative $\frac{d()}{d()}$

Before the sideways version, name its two symbols.

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7. The three star quantities (previews)

You now have every letter needed to read the parent note. Here are the three headline symbols, all already introduced above, gathered for reference:

Symbol	Name	Plain meaning	Units
$p$	vapour pressure	push of escaped gas at equilibrium	Pa or atm
$\gamma$	surface tension	pull per length of surface = energy per area	N/m = J/m²
$\eta$	viscosity	resistance to flow (internal friction)	Pa·s (or poise)

Recall Quick self-quiz on units

Surface tension $\gamma$ can be written as force per length OR energy per area — are those the same unit? ::: Yes: $\text{N/m}=\text{N⋅m}/{\text{m}}^2={\text{J/m}}^2$. Identical. One pascal-second equals how many poise? ::: $1\text{Pa⋅s}=10$ poise. What are the units of density $\rho$ and gravitational acceleration $g$? ::: $\rho$ in kg·m⁻³; $g\approx 9.81$ m·s⁻².

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Prerequisite map

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Equipment checklist

Test yourself — reveal only after answering.

• What are the two competing quantities behind every property in this chapter? ::: Intermolecular forces (stickiness) vs thermal energy (jiggle).
• Why must $T$ be in kelvin in these formulas? ::: Because $T$ sits in denominators/exponents; only kelvin's true zero keeps $1/T$ and ratios meaningful.
• What does $\Delta$ mean, and is $\Delta H_{vap}$ positive or negative? ::: "Change in" (final − initial); $\Delta H_{vap}$ is positive — you must supply energy to vaporise.
• What question does $\ln$ answer, and what does it undo? ::: "$e$ to what power gives this?"; it undoes the exponential $e^x$.
• Why does an exponential $e^{-E/RT}$ appear whenever molecules must beat a barrier? ::: It is the Boltzmann fraction of molecules whose energy $E$ clears that barrier.
• What does $\Delta G=0$ signify, and why does the vapour-pressure derivation start there? ::: Equilibrium — no net direction; liquid ⇌ vapour balance is where vapour pressure is defined.
• Geometrically, what is $\frac{dp}{dT}$? ::: The slope (steepness) of the boiling curve — how fast pressure rises per unit temperature.
• In the velocity gradient $\frac{du}{dz}$, what do $u$ and $z$ mean? ::: $u$ = a layer's flow speed; $z$ = position measured across the flow (slow layer to fast layer).
• What are the units of density $\rho$ and gravitational acceleration $g$? ::: $\rho$: kg·m⁻³; $g$: m·s⁻² (≈9.81).
• In $\eta =A{e}^{E_a/RT}$, what is $A$ and its units? ::: The pre-exponential ("try rate") factor setting baseline viscosity; units Pa·s.
• Why does $\cos \theta$ appear in the capillary-rise formula? ::: Only the vertical share of the tilted surface-tension pull holds the column up, and $\cos \theta$ extracts that vertical part.
• What sign does $\cos \theta$ take for mercury, and what happens at $\theta =90^{\circ }$? ::: For mercury $\theta >90^{\circ }$ so $\cos \theta <0\Rightarrow h<0$ (depression); at $\theta =90^{\circ }$, $\cos \theta =0\Rightarrow h=0$ (no rise or fall).
• Is $\gamma$ in N/m the same as J/m²? ::: Yes, identical units.

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Connections

• Parent topic ↗
• Intermolecular Forces — the shared root cause
• Boltzmann Distribution — origin of the exponential energy spread
• Gibbs Free Energy — why equilibrium means $\Delta G=0$
• Clausius-Clapeyron Equation — where these symbols combine for vapour pressure
• Capillary Action and Contact Angle — the geometry of $\theta$ and rise
• Boiling Point and Phase Diagrams — where vapour pressure meets $p_{\text{external}}$